New View of Motion under Gravitation without Classical Mysteries

Mysteries of Classical Mechanics

Gravitation force created by instant action at distance from local presence of matter?

Motion of matter (Zeno's paradox: moving arrow is still at each instant)?

Resolution of Mysteries by New View of Motion under Gravitation

Both mysteries can be resolved, or circumvented, by viewing the gravitational potential $\phi (x,t)$ depending on a Euclidean space coordinate $x$ and time $t$ as the primordial object, from which matter with density $\rho (x,t)$ and motion along trajectories $x(t)$ with acceleration $\ddot x(t)$ and velocity $\dot x(t)$ with the dot denoting differentiation with respect to time, are derived by the following equations to be satisfied for all $x$ and $t$:

$\rho (x,t) =\Delta\phi (x,t)$ (Newton's law of gravitation)

$\ddot x(t)= -\nabla\phi (x(t),t)$ (Newton's 2nd law).

The gravitational potential $\phi (x,t)$ itself is governed by the evolution equation

$\Delta\dot\phi = - \nabla\cdot (u\Delta\phi )$ (E)

where $u(x,t)$ is trajectory velocity defined by $u(x(t),t)=\dot x(t)$, which is a reformulation of mass conservation commonly expressed in the form

$\dot\rho +\nabla\cdot (u\rho )=0$. (mass conservation)

With the gravitational potential $\phi (x,t)$ as the primordial object, matter $\rho (x,t)$ will be created locally by the operation of the Laplacian $\rho (x,t) =\Delta\phi (x,t)$ expressing Newton's law of gravitation and trajectories $x(t)$ will be defined by Newton's 2nd law, and the gravitational potential will evolve governed by the evolution equation (E) which is a form of wave equation.

Wave Motion

The gravitational potential $\phi (x,t)$ will thus transform as a wave satisfying a wave equation and matter with density $\rho (x,t)$ will be created/annihilated locally according $\rho (x,t)=\Delta\phi (x,t)$

and thereby appear to follow trajectories $x(t)$ given by Newton's 2nd law.

Motion of matter will then be an illusion of real gravitational potential wave motion, in the same way that the illusion of horisontal motion of heaps of water across a water surface is generated by water molecules in reality in circular motion up and down, where heaps of water are continuously created/annihilated giving the illusion of progression:

Notice that the trajectories $x(t)$ are mathematical constructs, and not "real particle trajectories" since there are no "particles in motion along particle trajectories". The motion consists wave motion of the gravitational potential, which is geared to follow mathematical trajectories.

The Hen and the Egg

The gravitational potential $\phi$ thus appears as a hens farm capable of laying eggs (creating matter) anywhere which are picked up by the farmer (annihilation of matter). An illusion of eggs in motion can then be formed this way, while the reality is that each egg (like Zeno's arrow) lays still after being laid and before being picked up and a moved egg in reality is a new recreated egg.

Summary of New View

We are thus led to a New View of Motion under Gravitation offering a resolution of the main mysteries of classical mechanics of gravitational force acting at distance and Zeno's paradox of a moving arrow which is still and not moving at instant of time.

In the New View the primordial entity is the gravitational potential $\phi (x,t)$ governed by an evolution equation expressing mass conservation under motion. In the New View all action is local and Zeno's arrow is allowed to be still at each moment of time, yet capable of giving the illusion of motion by a process of creation/annihilation resulting from real wave motion of the gravitational potential $\phi$.

In particular in the New View there is no need to introduce gravitons as some form of "force carrying particles" transmitting gravitational forces over long distance by moving with infinite or finite speed. There is no evidence of the existence of such particles.

In the New View the only motion is wave motion in the gravitational potential with velocity given by trajectories governed by Newton's 2nd law.

In the New View the equations are those of classical Newtonian mechanics, but the interpretation and cause-effect is different. In the New View the gravitational potential is the cause, matter is the effect and motion of matter an illusion.

In the New View there is no need of general relativity with motion of matter governed by the "fabric of curved space-time", which is also a mystery understood by nobody. In the New View there is no need to speculate about the existence of gravitational waves progressing at the speed of light, which lack any form of experimental support despite major efforts.

The New View offers a look behind the scene of the visible motion on the stage of our planetary system with the planets accelerating towards the Sun as if governed by a God's hand of instant action at distance. What we can see on scene is mostly illusion of reality and not reality.

Is there then no mystery in the New View? Yes, of course there is and it is the generation of matter from gravitational potential according to $\rho =\Delta\phi$. This is the mystery of how a hen can generate an egg. We compare with the mystery of how an egg can generate a hen. Which mystery would you prefer to explain: (i) an egg from a hen or (ii) a hen from an egg? My choice is (i) and yours? To remove all mystery may well be beyond human capacity, but we can always seek to minimize the amount of mystery, or if your are a professional physicist to maximize the mystery in order to maximize government grants to support multiplication of string and multiverse theorists.

Notice that physicists (and philosophers) are still struggling to resolve Zeno's paradox without any success declared. Often a reference to Calculus as a resolution is made with a hint that the arrow appearing at different places at different times follows a differentiable curve with a certain momentary velocity defined by the limit process of differentiation, and so must be moving. But that is a fake solution because the essence which is the physics of an arrow appearing at different places at different times, is hidden and the limit process is purely mathematical. Thus Zeno's paradox of the impossibility of motion is unresolved in classical Newtonian mechanics, which does not reveal the physics of motion simply taking it for granted by pointing to the Moon traversing the sky.

The answer is correct, but it carries the counter-intuitive implication that motion is not something that happens at any instant, but rather only over finite periods of time. Think about it this way: time, as we said, is composed only of instants.

No distance is traveled during any instant. So when does the arrow actually move? How does it get from one place to another at a later moment?

There's only one answer: the arrow gets from point X at time 1 to point Y at time 2 simply in virtue of being at successive intermediate points at successive intermediate times—the arrow never changes its position during an instant but only over intervals composed of instants, by the occupation of different positions at different times.

4 kommentarer:

The obvious reply from me is that a moving object IS moving, at every "instant" (however such an instant is defined)--why constrain your thinking with such idiocy as "no distance is traveled during any instant"?--the object travels a distance in proportion to the size of the "instant" being considered, that is the obvious PHYSICAL statement of fact, and the whole idea of a paradox is simply a confusion of words, without understanding, from the perverse assumption of "no motion during an instant". Zeno was a fool, and any "physicist" (or "mathematician") who feels confused by his "paradox" should not be teaching the subject.

Nonsense, the argument is crystal clear. But you know as well as I do that the arrow reaches its destination, and therefore that Zeno's argument is in error. I have stated my scientific opinion, that the error obviously lies in a clearly false assumption, and you have responded with empty denial--why did you do that? I am here to inform those who will listen, not to argue fruitlessly with dogmatic beliefs. I reject Zeno, scientifically, on the clear grounds I have given above, and I think Zeno was a fool for not identifying the obviously faulty assumption as faulty--his foolishness continues with you and many others to this day, and I find that abhorrent. And that is my final word on this subject here.

Yes, the arrow reaches its destination and that is not a triviality as you seem to think.That you so quickly reach the bottom line of final word indicates that you are locked into a rest position without possibility of motion.